October 13, 2013

This is a list of elementary problems I like for one reason or another from various branches of mathematics. I cited the people I heard the problems from, they are not necessarily the originators. The unsolved flag just means that the problem is yet unsolved by me. Send me a solution if you find one. You can send me any interesting problems you find by adding a comment to this post. Also check out my other math posts, especially probability twisters, unsolved elementary problems, and links to other math sites. Last update: October 13, 2013.

(Quanta Magazine) Show that if Alice tosses a fair coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses.

(Cihan Baran) If we sample k times with replacement from the set {1, 2, ..., n} (all members picked with equal probability) what is the probability that at least one member will not be picked?

(Bertsekas and Tsitsiklis) Suppose that n people throw their hats in a box and then each picks one hat at random. (Each hat can be picked by only one person, and each assignment of hats to persons is equally likely.) What is the expected value of X, the number of people that get back their own hat?

(Paul Lockhart) Must there always be two people at a party who have the same number of friends there?

(Volkan Cirik) Consider a game played with an array of 2n random numbers. The first player can pick the number at either end of the array, the second player can pick from either end of the remaining numbers etc. The players take turns picking numbers from either end until no more numbers remain. Whoever has the highest total wins. Show that the first player can always win or draw.

(Ahmed Roman) Nathan and Abi are playing a game. Abi always goes first. The players take turns changing a positive integer to a smaller one and then passing the smaller number back to their opponent. On each move, a player may either subtract one from the integer or halve it, rounding down if necessary. Thus, from 28 the legal moves are to 27 or to 14; from 27, the legal moves are to 26 or to 13. The game ends when the integer reaches 0. The player who makes the last move wins. For example, if the starting integer is 15, Abi might move to 7, Nathan to 6, Abi to 3, Nathan to 2, Abi to 1, and now Nathan moves to 0 and wins. (However, in this sample game Abi could have played better!)

Assuming both Nathan and Abi play according to the best possible strategy, who will win if the starting integer is 1000? 2000? Prove your answer.

As you might expect, for some starting integers Abi will win and for others Nathan will win. If we pick a starting integer at random from all the integers from 1 to n inclusive, we can consider the probability of Nathan winning. This probability will fluctuate as n increases, but what is its limit as n tends to infinity? Prove your answer.

(Serdar Tasiran) We have an n boxes, k of them have a ball in them (k<=n), the others are empty. We start opening the boxes in a random order and stop when we find a ball. What is the expected number of boxes we will open?

(Ertem Esiner) Two mathematicians meet an old friend who has two kids and ask her what their ages are. The mother writes the sum of the two ages on a piece of paper and gives it to the first mathematician, and gives the product of the two ages to the second mathematician. The mathematicians think for a minute and claim the information is not sufficient. The woman says "think again". At that moment the mathematician with the product figures out the answer. How old are the two kids?

(Ertem Esiner) Five pirates are trying to divide up 1000 gold pieces among themselves. They decide to take turns making offers. An offer by a pirate is accepted if at least half of the other pirates agree to it. Otherwise the pirate making the offer is killed, and the next one makes an offer. Each pirate is greedy, but none wants to die. If you are the first pirate, what offer do you make?

(Drake) A professor announces that there will be surprise exam the following week, specifically that the students will not know what day the exam is going to take place. The students reason that the exam cannot take place on Friday, because by Thursday night they would know what day the exam is going to take place. If the exam cannot be on Friday it cannot be on Thursday either, because they would know by Wednesday night and so on. They finally decide the exam cannot take place and forget to study. The professor gives the exam on Wednesday to everybody's surprise. What went wrong?

(Mackay) Fred rolls an unbiased six-sided die once per second, noting the occasions when the outcome is a six.

What is the mean number of rolls from one six to the next six?

Between two rolls, the clock strikes one. What is the mean number of rolls until the next six?

Now think back before the clock struck. What is the mean number of rolls, going back in time, until the most recent six?

What is the mean number of rolls from the six before the clock struck to the next six?

Is your first answer different from your last answer? Explain.

(Deniz) You have two independent random variables between 0 and 1. How do you decide which one is more likely to be larger than the other?

(Deniz) You have two arbitrary random variables between 0 and 1. How do you decide if they are independent or not looking at their joint pdf density plot?

(Dennis Eriksson) Find all solutions to the diophantine equation: 1+2+3+...+n=m^2, where n and m are positive integers.

(Feyz) You have a deck of n cards numbered from 1 to n. dealt and shuffled randomly. What is the probability that none of the i-th card is on the i-th position?

(Sonny) Prove that there is a natural number n, for which 2^n starts with the numbers 3141592, i.e., show that there is a number of the form 3141592.... which is a power of 2 (in base 10 representation).

(Alkan) Let n, k be integers greater than 1.

Show that 1/1 + 1/2 + 1/3 +...+ 1/n cannot be an integer.

Show that 1/k + 1/(k+1) + ... + 1/(k+n) cannot be an integer.

(Will,Minsky) These problems have something in common:

A monk leaves to ascend to the temple on top of a mountain at 9am and arrives at 5pm. The next day he leaves the temple at 9am and arrives back at the foot of the mountain at 5pm. Prove that there is a point in time where he was at the same location on the path at the same time.

Prove that on a 2D earth, there exists a diameter such that the temperature at the endpoints is equal.

Prove that on a 3D earth, there exists a diameter such that the temperature and humidity of the endpoints are equal.

Does every convex closed curve in the plane contain all four vertices of some square?

(Ben) Two nice algorithm questions: Given a shuffled array of numbers from 1 to 10,000, find the three that are missing in one pass. Given an array of positive and negative integers, find the subarray with the highest sum in linear time.

(Winston) Four people want to pass a bridge dark at night. They can walk the bridge in 1, 2, 9, and 10 minutes respectively. The bridge can carry at most two people at a time. There is a single flash-light, and they need the flash-light to walk on the bridge. What is the shortest time for all four to pass across? (This was apparently a popular Microsoft interview question).

(Beril) You have a glass of tea and a glass of milk. You take a spoonfull of milk, mix it with the tea. Then you take a spoonfull of this mixture and mix it with the milk. Is there more milk in the tea or more tea in the milk at the end?

(Mine) Construct a square from: (a) Three identical squares, (b) Any two squares, (c) A rectangle. (d) Divide a square into any given two squares with the same total area. (e) Divide a circle into 6, 7, 8, and 10 equal pie-slices.

(IMO practice) m+n people are standing on a movie line. m people have 5 dollar bills, n people have 10 dollar bills. The movie is 5 dollars. The cashier opens with no money. It will close if it does not have enough change to give one person. How many possible lines are there that will get through without closing the cashier?
Note: (Deniz, Mar 10, 1998) I just discovered that this problem is equivalent to finding the number of full binary trees with m leaves when n=m-1. A full binary tree is a tree where each node has 0 or 2 children. The number of binary trees is equivalent to the number of shift-reduce sequences that parse them. For such a sequence to be valid the number of shifts need to always be ahead of the number of reduces, which turns this into our movie problem. The binary tree problem can also be solved using a generating function and the relation b[n] = sum[k=1..n-1](b[k] b[n-k]). The movie problem can be solved by using random walks and the reflection principle. The two solutions seem to give different answers but they turn out to be equivalent. This constitutes an indirect proof of the following combinatorial identity: (2n-1)!! = 2n!/(n! 2^n). Everything is related to everything else in math :-)

(Oguz) Find a function f on real numbers such that f(f(x)) = -x.

(Boris) You meet many women in your life. After meeting each one, you decide how good she is and whether you want to marry her. If you decide to marry her, you lose your chance with future candidates. If you decide to move on, you lose your chance with her. Assuming you will meet at most n women, find the optimum strategy for marrying the best bride.
(The Azeri mathematician Gussein-Zade is apparently the first one to solve this problem.)

(Alkan) A small rectangle is cut out of a large rectangle. Find a line that divides the remaining figure into two equal areas using an unmarked ruler.

(IMO) Let A be a set of ten two-digit integers. Prove that one can always find two subsets of A with the same sum.

(IMO) 17 people correspond with each other. Each pair discusses one of three possible topics. Prove that there are three people that discuss the same topic with each other.

(Alkan) Five couples meet in a party. Everyone starts shaking hands with everyone else except their partners. At some point the host stops them and asks how many handshakes each had. Everyone gives a different number. How many hands did the host's wife shake?

(IMO-75/4) When 4444 4444 is written in decimal notation, the sum of its digits is A. Let B be the sum of the digits of A. Find the sum of the digits of B. (A and B are written in decimal notation.)

(Murat Fadiloglu) What is the probability of two randomly selected integers being mutually prime?

(Alkan) An old lady buys a plane ticket to visit her son. She goes to the airport and people let her board first. Since she can't read her seat number, she sits on a random seat. Rest of the passengers sit on their own seats, unless it is occupied in which case they randomly choose one of the emtpy seats. What is the probability that the last passenger will sit on his own seat?

(Rota) Given a sequence of (n 2 + 1) distinct integers, show that it is possible to find a sequence of (n+1) entries which is increasing or decreasing.

(Minkowsky) Consider a two dimensional lattice with grid points unit distance apart. Show that a convex shape that is symmetric around a grid point has to include at least three grid points if its area is 4.

(Science Museum, unsolved) Which rectangles can be divided into unequal squares?

(Alkan) Consider permutations of an array which contains n copies of each integer from 1 to n. Two permutations are defined as orthogonal if their corresponding elements form distinct pairs. What is the maximum number of permutations such that any two are orthogonal? For example, here is a possible set of mutually orthogonal permutations for n=3: {111222333, 123123123, 123231312, 123312231}.

(Ian) My new favorite algorithm: Find an algorithm that discovers if a linked list has a cycle in it using bounded memory.

(Uttrash) You are in prison and they give you n red balls, n green balls and two boxes. You are to place the balls in the two boxes in any way you like. The next day they will pick a ball from one of the boxes, and if it is green you will be set free. How do you arrange the balls?

(Will) You randomly throw k balls into n bins. What is the expected number of occupied bins.

(Will) You randomly throw k points on the unit interval. What is the expected length of the longest segment.

(Michael, unsolved) You distribute 100 balls to 10 buckets. What is the expected value of the number of balls in the bucket with most balls.

(Lon) Draw 2 circles, 1 completely inside the other (but not necessarily concentric.) What is the probablility that a line intersecting the outer circle also intersects the inner circle. Now, do the same with rectangles.

(Thurston and Conway) An angel is stuck on an infinite sheet of graph paper, he can hop from square to adjacent square. Everytime the angel hops, the devil can knock out any square, so the angel can't ever go there. Can the devil trap the angel? What if the graph paper is a positive quadrant (i.e. is bounded on two sides).

This is not really math, but here are my two favorite algorithms: (1) Find an algorithm for perfect shuffling of an array. (2) Find an algorithm that will pick a perfectly random element from a list in one pass without knowing the size of the list beforehand.

(Alkan) Given two points find the point midway between them using only a compass (no ruler).

(Alkan) You are sitting at point [0,0] and looking towards right into a tunnel bounded by y=1/x and y=-1/x curves. The walls of the tunnel are reflecting. Prove that if you send a light beam into the tunnel in any direction other than straight to the right, the beam will be reflected back towards left.

(Deniz) Let x be a random variable which can take positive integer values. P(x)=1/2x. We draw n random elements from this distribution. What is the probability that the n+1st element will be different from the first n?

(Alkan) Let A be the set of all rectangles that have one integer side. Prove that any rectangle constructed by concatenating rectangles from A will also be a member of A.

(Neal) Take a randomly shuffled deck. Open the cards one by one. At one point stop and predict that the next card is red. Is there a strategy that has more than 1/2 chance.

Pick two random points in the unit line segment. What is the expected distance between them?

Pick two random points in the unit circle. What is the expected distance between them?

(Umit) Suspend a rope from two points on the ceiling. What shape does it take?

(Bernoulli brothers) A ball is rolling from point A to a lower point B. What is the ideal curve for the path between A and B that minimize the travel time?

(Alkan) There are 100 light poles with switches on a street. In the beginning all lights are off. One person goes through pole number 1, 2, 3, ... and flips the switches. Then he goes back and goes through 2, 4, 6, ... and flips the switches. Then he goes back and goes through 3, 6, 9, ... and flips the switches. So at n'th round he flips the multiples of n. Which lights are on after 100 rounds?

(Deniz) There is a set A of n0 elements, and we randomly pick a subset B of n1 elements. We know that r0 of the elements in A were red. We are interested in finding out the number of red elements in B, r1. To find out we start picking random elements from B. We pick n2 elements, and r2 of them turn out to be red. Now what is the best estimate for r1?

(Minsky) I bring you three flipped cups and tell you there is gold under one of them. Furthermore, each cup has a number giving the probability that the gold is under that one. You immediately go to the one with highest probability. I tell you that you have amnesia and I may have tried this on you a million times. What is your best strategy?

(Minsky) An ant leaves a repeated binary pattern behind as it walks through the desert. What is the length of the shortest pattern that would let you distinguish which way the ant was going?

(Feyzu) Two points are chosen at random on a line AB, each point being chosen according to the uniform distribution on AB, and the choices being made independently of each other. The line AB may now be regarded as divided into three parts. What is the probability that they may be made into a triangle?

(IMO practice) The entries for a competition is locked in a safe. There are 11 judges. We would like them to be able to open the safe when more than half get together. How many locks / keys do we need?

(IMO practice) Given three parallel lines, show that an equilateral triangle can always be constructed with a vertex on each line.

(IMO-72/6) Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

(Umit) A method for two people to share a pie fairly is to let one cut the other one pick. Generalize this method to n people.

You are making a random walk on an n-dimensional grid. What is the probability that you will ever return to the origin? (Hint: It is 1 for 1-D and 2-D! It is 0.3405 for 3-D).

(Ivanie) A rabbit hopping up the stairs can hop either one or two steps at a time. How many different ways can it climb n steps?

Show that if you cut off two opposite corner squares of a chess board, you cannot cover the rest with dominoes.

Show that n squares with total area less than 1/2 can always be fit into a unit square (non-overlapping).

Show that n squares with total area greater than 3 can always cover the surface of the unit square (non-overlapping).

You color all points of a plain with three colors. Show that I can always find two points of the same color that are a given distance apart.

You color all points on an equilateral triangle with two colors. I try to find a right triangle with its vertices on the edges of your triangle and all vertices having the same color. Can you find a coloring that prevents this?

How many 1's are there in the digits of numbers from 1 to 1 million? (one minute time limit).

(Michael) There is a piece of candy on every node of a binary tree. Find the shortest path through the binary tree that lets you collect all of the candies.

(Ihsan) Two men, x distance apart, start walking toward each other with speed v. At that instant a fly starts flying from one men's nose to the other with 2v speed. The fly keeps flying back and forth between the two noses until the guys meet. How much distance has the fly flown when they meet? (There is an easy way and a hard way to solve this).

(Ihsan) A coin is flipped until the first head appears. If you get a head in n flips you win $2n. How much are you willing to pay to play this game?

(Bilim ve Teknik) You need to paint the area under the curve 1/x. How can you do it with a finite amount of paint?